Question

Put the differential equation 9ty+ety′=yt2+81 into the form y′+p(t)y=g(t) and find p(t) and g(t). p(t)= help (formulas) g(t)= help (formulas) Is the differential equation 9ty+ety′=yt2+81 linear and homogeneous, linear and nonhomogeneous, or nonlinear?

Answer 1

`p(t) = (9t^(3) + 729t - 1)/(e^(t)(t^(2) + 81) )`

g(t) = 0

And

The differential equation
`9ty + e^(t)y' = (y)/(t^(2) + 81 )` is linear and homogeneous

Explanation:

Given that,

The differential equation is -

`9ty + e^(t)y' = (y)/(t^(2) + 81 )`

`e^(t)y' + (9t - (1)/(t^(2) + 81 ) )y = 0\\e^(t)y' + ((9t(t^(2) + 81 ) - 1)/(t^(2) + 81 ) )y = 0\\e^(t)y' + ((9t^(3) + 729t - 1)/(t^(2) + 81 ) )y = 0\\y' + [(9t^(3) + 729t - 1)/(e^(t)(t^(2) + 81) ) ]y = 0`

By comparing with y′+p(t)y=g(t), we get

`p(t) = (9t^(3) + 729t - 1)/(e^(t)(t^(2) + 81) )`

g(t) = 0

And

The differential equation
`9ty + e^(t)y' = (y)/(t^(2) + 81 )` is linear and homogeneous.